The set of real numbers contains infinitely many terms, each one greater than the last.

How can there be a supremum?

Surely the set has infinitely many terms, each one greater than the last. Clearly has supremum though.

When I learned about supremums we were taught the convention that an empty set has supremum and a set unbounded above has supremum , so the set of real numbers has supremum , this is known as an affine extension.